Find the smallest number which when divided by 4, 9, 12 and 16 always leaves the remainder 3.

Factors of 16 are numbers that, when multiplied in pairs give the product as 16. There are 5 factors of 16, which are 1, 2, 4, 8, and 16. Here, 16 is the biggest factor. The Prime Factors of 16 are 1, 2, 4, 8, 16 and its Factors in Pairs are (1, 16), (2, 8), and (4, 4).

• Factors of 16: 1, 2, 4, 8 and 16
• Negative Factors of 16: -1, -2, -4, -8 and -16
• Prime Factors of 16: 2
• Prime Factorization of 16: 2 × 2 × 2 × 2 = 24
• Sum of Factors of 16: 31

What are the Factors of 16?

A number that divides another number without leaving any remainder is called the factor of that number. When we divide 16 by 2, it is exactly divisible and leaves no remainder. Therefore, 2 is a factor of 16. Similarly, 4 and 8 leave no remainder and hence, they are factors of 16.

Note that 1 and the number are always factors of the number. Factors of 16 are 1, 2, 4, 8, and 16.

How to Calculate the Factors of 16?

Prime factorization or the division method can be used to find the factors of 16. Using the division method, we can find the numbers which divide 16 exactly without leaving a remainder.

Factors of 16 by Division Method

• For finding the factors of 16 by division method, we will divide 16 by counting numbers and see which numbers exactly divide 16 and give 0 as a remainder.
• For example, when we divide 16 by 2 or 4 or 8, the remainder will be 0.
• At the same time, when we divide 16 by numbers like 3 or 5 or 7, it leaves a remainder.
• Let us divide and check for all numbers.

Thus, all numbers which leave a remainder 0 are the factors of 16. We see that 1, 2, 4, 8, and 16 are the factors of 16. Explore factors using illustrations and interactive examples:

• Factors of 162- The factors of 162 are 1, 2, 3, 6, 9, 18, 27, 54, 81, and 162.
• Factors of 169- The factors of 169 are 1, 13, and 169.
• Factors of 216- The factors of 216 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, and 216.
• Factors of 160- The factors of 160 are 1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 80 and 160.
• Factors of 168 - The factors of 168 are 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, and 168.

Factors of 16 by Prime Factorization

This is done by dividing 16 by prime numbers. We will then check if it does not leave any remainder and continue division with the quotient if it is a composite number. We can find the prime factors of 16 by the division method or the factor tree method.

Prime Factorization of 16 by Division Method

• Step 1: The number 16 is divided by the smallest prime number which divides 16 exactly.
• Step 2: The quotient thus obtained is then divided by the smallest or second smallest prime number and the process continues till the quotient cannot be further divided.
• Step 3: Let us divide 16 by the prime number 2 i.e. 16 ÷ 2 = 8
• Step 4: Now we divide 8 by 2 and repeat the process until the quotient cannot be further divided.

Prime Factorization of 16 by the Factor Tree Method

Here we are finding the prime factors of 16, therefore, the root of our factor tree is 16. We write the pair of factors as the branch of 16.

2 is a prime number. Hence, the factor tree ends there. Hence, prime factorization of 16 is 2 × 2 × 2 × 2

Prime Factorization of 16 by Upside Down Division Method

This method is called the upside-down division because the division symbol is flipped upside down. Here we divide 16 by the smallest prime number 2. If the given number is odd, we divide it by 3. We will divide till we get 1.

By this we conclude that 16 = 2 × 2 × 2 × 2.

Think Tank:

• Can we write the prime factorization of 16 as 24?
• Which is the biggest factor of 16?
• Is 16 a square number? Why?

Factors of 16 in Pairs

A factor pair is the set of two numbers that when multiplied will give the number as the product. Factor pairs of 16 are:

• 1 × 16 = 16
• 2 × 8 = 16
• 4 × 4 = 16

Therefore, the factor pairs of 16 are (1, 16), (2, 8), and (4, 4).

Since the product of two negative numbers is positive, the negative factor pairs of 16 are (-2, -8), (-4, -4), and (-1, -16).

Prime Factors of 16 By Division Method

For the division method, 16 should be divided by the smallest prime number which divides it exactly without leaving any remainder.

• Step 1: If 16 is divided by 2, the quotient will be 8, 16 ÷ 2 = 8
• Step 2: The number 8 is not a prime number; therefore, it is further divided.
• Step 3: If 8 is divided by 2, the quotient will be 4 i.e. 8 ÷ 2 = 4
• Step 4: The number 4 is not a prime number. Hence, it can be further divided as 4 ÷ 2 = 2.
• Step 5: 2 is a prime number. Thus, it cannot be further divided.

Therefore, the factors of 16 are 1, 2, 4, 8, and 16 and the prime factorization of 16 is 16 = 2 × 2 × 2 × 2.

Important Notes:

• Factors of 16 are 1, 2, 4, 8, and 16.
• 1 is a universal factor.
• The two numbers that are multiplied to get a product are the factors of the product.

1. Example 1: Can you help Tom list the factors of 24 and check if the numbers 24 and 16 have any common factors?

   Solution:

   There are eight factors of 24:

   24 ÷ 1 = 24

   24 ÷ 2 = 12

   24 ÷ 3 = 8

   24 ÷ 4 = 6

   24 ÷ 6 = 4

   24 ÷ 8 = 3

   24 ÷ 12 = 2

   This means that the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24 and the factors of 16 are 1, 2, 4, 8, and 16. Therefore, 24 and 16 have 1, 2, 4, and 8 as common factors.

2. Example 2:

   Amy wanted to identify all the positive factor pairs of 16. Can you help her?

   Solution:

   Factor pairs of 16 are:

   1 × 16 = 16

   2 × 8 = 16

   4 × 4 = 16

   Therefore, the positive factor pairs of 16 are (1,16), (2,8), and (4,4).

3. Example 3: Find the product of all the prime factors of 16.

   Solution:

   Since the prime factors of 16 are 2. Therefore, the product of prime factors = 2 = 2.

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The factors of 16 are 1, 2, 4, 8, 16 and its negative factors are -1, -2, -4, -8, -16.

What Numbers are the Prime Factors of 16?

The prime factor of 16 is 2.

What is the Greatest Common Factor of 16 and 5?

The factors of 16 are 1, 2, 4, 8, 16 and the factors of 5 are 1, 5. 16 and 5 have only one common factor which is 1. This implies that 16 and 5 are co-prime.

Hence, the Greatest Common Factor (GCF) of 16 and 5 is 1.

What is the Sum of all the Factors of 16?

Factors of 16 are 1, 2, 4, 8, 16 and, the sum of all these factors is 1 + 2 + 4 + 8 + 16 = 31

How Many Factors of 15 are also common to the Factors of 16?

Since the factors of 16 are 1, 2, 4, 8, 16, and factors of 15 are 1, 3, 5, 15. Hence, 16 and 15 have only one common factor which is 1. Therefore, 16 and 15 are co-prime.

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